H-Infinity Control Based Scheduler for the Deployment

of Small Cell Networks

Subhash Lakshminarayana,1,2,∗, Mohamad Assaad1,∗, Merouane Debbah2,∗

Abstract

In this work, we address the joint problem of traffic scheduling and interference

management related to the deployment of Small Cell Networks (SCNs). The

base stations of the SCNs (which we will refer to as Micro Base Stations, MBSs)

are low power devices with limited buffer size. They are connected to a Central

Scheduler (CS) with limited capacity backhaul links. In this scenario, traffic

has to be scheduled from the network to the MBS queues in such a way that the

queue-length at MBS remains as close as possible to a given target queue-length.

The challenge is to design a scheduler which is oblivious to the wireless link be-

tween the MBSs and the User Terminals (UTs). For the traffic arriving at the

MBS, we need to efficiently transmit it over the wireless channel to the UTs in

an interference limited environment. Additionally, real time centralized inter-

ference management techniques will not be feasible. In this paper, we decouple

the joint scheduling and interference management into two separate parts. For

the scheduling problem, we propose a H∞ control based scheduler which regu-

lates the arrival rates to the queues at the MBS. For the problem of interference

management over the wireless channel, we propose a multi-cell beamforming

technique and formulate a decentralized algorithm using tools from the field

∗Corresponding author.Email addresses: subhash.lakshminarayana@supelec.fr (Subhash Lakshminarayana,),

mohamad.assaad@supelec.fr (Mohamad Assaad), merouane.debbah@supelec.fr (MerouaneDebbah)

1Department of Telecommunications, SUPELEC, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette,France

2Alcatel-Lucent Chair on Flexible Radio, SUPELEC, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette, France

Preprint submitted to Performance Evaluation September 2, 2011

of random matrix theory. The beamforming vectors are designed to optimize

two performance metrics of interest namely downlink power minimization and

weighted sum rate maximization. Our simulation results show that the H∞

based queue length control algorithm stabilizes the queue-lengths at the MBS

and keeps the variation of the queue-length around the target to a minimum.

Keywords: Small cell netwoks, Random matrix theory, Scheduler, Beam-

forming, H∞ control

1. Introduction

According to the observation made by Martin Cooper of Arraycomm, during

the last century, the capacity of wireless networks has doubled every 30 months.

It has been observed that the biggest gains in the efficiency of spectrum uti-

lization has been due to network densification, i.e., shrinking of cell sizes [1].

Hence the next paradigm shift in the field of cellular networks has been identi-

fied as shrinkage of cell sizes resulting in Small Cells Networks, (SCN) [2]. The

reduction in the cell sizes implies reducing the distance between the transmit-

ter and the receiver, creating the dual benefit of higher link qualities and more

spatial reuse. It also implies reduction in the amount of power transmitted. In

this document, we will refer to the SCN base stations as Micro Base Stations

(MBSs). Typically the MBSs of the small cells are of the form of data access

points with limited battery and buffer sizes installed by the users to provide

better data and voice coverage over a small area.

However, the reduction of cell sizes imposes numerous technical challenges

in the design of such SCNs. Some of the technical issues faced in the implemen-

tation of SCNs are the problem of interference management, traffic scheduling,

optimal cell size planning, call handover issues, security, backhaul infrastruc-

ture etc. For a detailed survey on the SCN deployment and design challenges

in SCNs, please refer to [2],[3],[4].

In this work, we address two of the issues associated with the deployment

SCNs namely interference management and traffic flow regulation.

2

CS

Q1

Q2

Q3

UT1

UT2

UT3

MBS1

MBS2

MBS3

Figure 1: System Model

Consider the scenario in Figure 1. The set up consists of small cells with

MBS serving the UTs over the wireless channel. The MBSs are equipped with

multiple antennas and the UTs have a single antenna each. Inter-cell interfer-

ence is a major bottleneck in achieving good data rates for the UTs, specially

the UTs at the cell edge. Hence the MBSs perform joint multi-cell beamforming

in order to manage interference. However, the MBSs must perform beamform-

ing in a decentralized manner. To this end, we use joint multi-cell beamforming

technique developed in [5] using tools from random matrix theory [6]. We con-

sider two performance metrics of interest namely downlink power minimization

and weighted sum rate maximization.

The MBSs are in turn connected to a Central Station (CS) which forwards

the traffic arriving from the backbone network to the MBSs through wired

backhaul link. We assume that the CS is an infinite reservoir of packets. Since

the buffer sizes at the MBS is limited, we need an efficient flow controller which

regulates the traffic from the CS to the MBS in such a way that the queue-length

at the MBS remains as close as possible to a given target queue-length (usually

the capacity of the buffer at the MBS). The main challenge in the design of the

3

flow controller is that it must regulate the traffic flow while being oblivious to

the wireless channel conditions between the MBS and the UTs. This is a very

practical assumption, since the CS is typically connected to a large number

of MBSs and hence the cannot keep track of the channel conditions of all the

MBSs.

We model the queue-length evolution as a linear dynamic system and de-

velop a robust queue-length controller/regulator based on H∞ control [7]. The

H∞ type robust controller controls the trajectory of a linear dynamical system

under unknown disturbances (noise) without the knowledge of the statistical

distribution of the noise process. The advantage of such a control algorithm in

our setting is that the MBS do not need to feedback the CSI of the UTs to the

CS. The task of the H∞ controller is to specify an instantaneous arrival rate

every time slot (from the CS to the queues present at the MBS) such that it

minimizes the variation of queue length size around the target queue length.

1.1. Comparison with Cross-Layer Optimization in Wireless Networks

The problem handled in this paper is multi-disciplinary and spans the fields

scheduling and power allocation in wireless networks and decentralized interfer-

ence management techniques in the context of cellular networks. The problem

set up described above has a flavor of cross-layer design to it. Cross-layer opti-

mization has been a well studied topic in the context of single-hop and multi-hop

wireless networks [8], [9], [10]. In these works, the problem framework naturally

leads to decomposition of the problem into two parts. A fairness based flow

controller to regulate the traffic flow into the queues and a max-weight based

scheduler to select a set of flows whose packets can be scheduled for transmission

at each time slot.

Our work is loosely connected to the cross layer framework of the above

works. In our work, the scheduling aspect of cross-layer design is replaced

by beamforming on the wireless channel. Equipped with multiple antennas,

the MBSs obtain spatial degrees of freedom for interference management. The

MBSs can simultaneously serve multiple UTs by beamforming on the wireless

4

channel. The flow controller part is replaced by an H∞ based flow controller.

However, unlike the fairness based congestion controllers in previous works, the

objective of the H∞ based flow controller is to minimize the variation of the

queue-length around a given target queue-length. The application ofH∞ control

for queue-length stabilization in the context of wireless network has been used in

[11],[12]. In particular, [12] addresses a dynamic resource allocation problem in

multi-service downlink OFDMA systems. However, to the authors’ knowledge,

the joint problem of scheduling using H∞ control theory, power allocation and

beamforming has not been handled before in the context of cellular networks.

Throughout this work, we use boldface lowercase and uppercase letters to

designate column vectors and matrices, respectively. For a matrix X, Xi,j

denotes the (i, j) entry of X. XT and XH denote the transpose and complex

conjugate transpose of matrix X. We denote an identity matrix of size M as IM

and diag(x1, ..., xM ) is a diagonal matrix of size M with the elements xi on its

main diagonal. We use x ∼ CN (m,R) to state that the vector x has a complex

Gaussian distribution with mean m and covariance matrix R. We will use the

notation (x)+ = max(x, 0). We will also use the notation ||x||2W to denote the

weighted second order norm of the vector x given by xHWx. We use E [Y ] to

denote the expected value of a random variable Y.

The rest of the paper is organized as follows. In section 2, we provide the sys-

tem model and introduce the notations. We introduce the problem formuation

in two scenarios namely downlink power minimization and weighted sum rate

maximization. In section 3, we provide the algorithm description for the decen-

tralized beamforming design in the two scenarios described above. In section

4, we provide the details of the flow control algorithm based on H∞ control.

In section 5, we compare the performance of the H∞ against standard LQG

based flow control. The simulation results are provided in section 6. We also

provide a brief description of the details of H∞ based control and LQG control

in Appendix A and B respectively.

5

2. System Model

The system model is shown in Figure 1. It can broken down into two parts.

The frontend consisting of the MBSs and the UTs and the backend consisting

of the CS and the backhaul links.

We focus on the frontend first. We consider a SCN scenario consisting N

cells and K UTs per cell. The UTs in each cell are served by their MBS which

is equipped with Nt antennas and each UT has a single antenna. The nota-

tion UTi,j denotes the j-th UT present in the i-th cell. The MBS of each

cell serves only the UTs present only in its cell. Let hti,j,k ∈ CNt denote the

channel from the MBS i to the UTk,j during the coherence interval t. We as-

sume that the elements of the channel vector are Gaussian distributed, i.e.,

hti,j,k ∼ CN (0, (σ2

i,j,k/Nt)INt), 3 the variance of which depends upon the path

loss model between BS i and UTj,k. The channel variance has been scaled by

the factor Nt to maintain the per antenna power constraint at each base station.

We denote the coherence interval of the channel by the notation Tc.

We consider that a discrete time system in which the time slots are indexed

by t. Before proceeding further, we will clarify the notion of time slot t in our

context. We assume that arrivals and departures from the queues present at the

MBS happen once every coherence interval of the channel. Hence the notation

t indexes the coherence periods of the channel.

The backend part consists of the CS and the backhaul links connecting the

CS to the MBS. The task of the CS is to schedule the traffic arriving from the

underlying wired backbone into the queues present at the MBS. We assume that

the backhaul links have just enough capacity to deliver all the traffic scheduled

by the CS during a given coherence.

Let us denote the queue length at MBS i for the UTi,j in its cell during the

time slot t by the notation qti,j . We denote the target buffer length at the MBS

3Note that the variance of the channel is constant across all time slots since we consider

that the UTs do not move.

6

by the notation qi,j .We denote the traffic arrival rate into the queue of the MBS

i for UTi,j during the time slot t by the notation ati,j . We also denote the rate

at which packets are transmitted out of the queue (i, j) into the wireless link,

(i.e. the service rate) during the time slot t by the notation µti,j .

We now describe the design parameters of the problem. Let wti,j ∈ CNt de-

note the transmit downlink beamforming vector for the UTi,j during time slot t.

Likewise, let Γti,j denote the received SINR for UTi,j and γti,j the corresponding

target SINR during the time slot t. The received signal yti,j ∈ C for the UTi,j

during the time slot t is given by

yti,j =

K∑

l=1

htH

i,i,jwti,lx

ti,l +

N∑

m=1m 6=i

K∑

n=1

htH

m,i,jwtm,nx

tm,n + zti,j

where xti,j ∈ C represents the information signal for the UTi,j during the time

slot t and zti,j ∼ CN (0, σ2) is the corresponding additive white Gaussian complex

noise.

We now provide two different problem formulations depending on the metric

of interest namely the downlink power minimization and the weighted sum rate

maximization.

Downlink Power Minimization - PMIN : The downlink power minimization

and scheduler design problem can be cast into the following optimization prob-

lem given by

PMIN : minwt

i,j

∑

i,j

wtH

i,jwti,j , ∀t = 1 . . . T (1)

s.t. Γti,j ≥ γti,j , t = 1 . . . T, i = 1 . . .N, j = 1 . . .K

limT→∞

1

T

T∑

t=1

qti,j = qi,j

The objective function of the optimization problem in (1) attempts to mini-

mize the total power transmitted in the downlink during each time slot t. The

first constraint equation is the target SINR constraint for each UT. The sec-

ond constraint equation denotes the scheduler design constraint which tries to

maintain the long term average of the queue-lengths (across multiple coherence

7

intervals) as close as possible to the target queue-lengths. The received SINR

in the downlink is given by the following expression

Γti,j =

|wtHi,jh

ti,i,j |

2

∑

l 6=j |wtHi,lh

ti,i,j |

2 +∑

m 6=i,n |wtHm,nh

tm,i,j |

2 + σ2(2)

The numerator term is the useful signal. The denominator terms represent the

intra-cell interference, inter-cell interference and the thermal noise (in order as

they appear in the denominator).

Weighted Sum Rate Maximization - RMAX : We now consider the problem

of maximizing the weighted sum rate subject to max-power constraint on each

MBS. The optimization problem is given by

RMAX : maxwt

i,j

∑

i,j

βi,j log(1 + Γti,j), ∀t = 1 . . . T (3)

s.t.∑

j

wtH

i,jwti,j ≤ Pmax, t = 1 . . . T, i = 1 . . .N

limT→∞

1

T

T∑

t=1

qti,j = qi,j

where the downlink SINR Γti,j is given as in equation (2). Here, βi,j are scalar

weights for the downlink rate of UTi,j and Pmax is the max-power constraint

per MBS.

3. Algorithm Design

The basic optimization problem in (1) and (3) are difficult to solve due to

the interdependencies of the various parameters involved. Hence, in this work

we decouple the problem into two stages. The first stage is the scheduling or

the rate control problem for the queues of the MBS. The design objective of the

scheduler is to minimize the variance of the queue-length around a given target

queue-length. The second stage is the interference management problem.

The parameters connecting the two problems are the SINR and the service

rates of the queues. The relationship between the service rate of the queues at

8

the MBS, µ and the SINR, γ, is given by the Shannon-Hartley theorem 4

µ = B log(1 + γ) (4)

where B is the total number of channel uses available during the coherence

interval Tc. Accordingly, we denote the service rate corresponding to the target

SINR (γi,j) for UTi,j by µi,j and the service rate corresponding to the achieved

SINR in the downlink (Γti,j) during each time slot t by µt

i,j . We will refer to µi,j

as the target service rate for the queues.

The decoupled algorithm can be summarized as follows:

• The UTs request a target SINR during each time slot depending on their

requirement.

• The beamformer design algorithm computes the optimal design parame-

ters to serve the data to the UTs.

• During each time slot, the scheduler regulates the amount of traffic arriv-

ing into the queues of the MBS such that the queues are stable.

In the subsequent part of this section, we will describe the decentralized multi-

cell beamforming algorithm in detail. We will defer the flow regulation part to

section 4. The problem of decentralized multicell beamforming has received con-

siderable attention in recent years [13], [14], [15], [16]. [13] suggests beamform-

ing design based on maximizing the virtual signal to interference and noise ratio

(VSINR) metric which is shown to attain optimal rate points. [14] proposes a

distributed beamforming approach based on Kalman smoothing involving mes-

sage passing between the BSs. [15], [16] propose decentralized beamforming

technique based on localized message passing, hence eliminating the need for a

central processor. The decentralized algorithm proposed in this work is based

on random matrix theory based approximations. In our algorithm, the MBSs

4We assume that the coherence interval of the channel is long enough for the transmitter

to achieve the channel capacity.

9

would need to exchange only the channel statistics between themselves and not

the instantaneous CSI, hence reducing the information exchange. Such algo-

rithms are asymptotically optimal (when the system dimensions become very

large, to be explained later in this section) and provide good approximations

for practical network dimensions. We now describe the algorithm in detail.

3.1. Beamformer Design and Power Allocation: The Downlink Power Mini-

mization Case

We first provide the a distributed algorithm to compute the optimal beam-

forming vectors and downlink power minimization problem in (1). We take

up the beamformer design problem during a given time slot t. Hence we drop

the superscript t in subsequent equations related to the above problem. The

optimization problem for the beamforming design is given by

minwi,j

∑

i,j

wHi,jwi,j (5)

s.t. Γi,j ≥ γi,j , i = 1 . . .N, j = 1 . . .K

The beamformer design problem is inspired from our previous work in [5].

The downlink power minimization problem in equation (5) is more difficult

to solve since the computation of the beamforming vector of a given UT in turn

depends on the beamforming vectors of the other UTs as well. Hence, using an

approach similar to the one in [17], we convert this problem into a dual uplink

problem. The uplink problem is much easier to be solved. The dual problem in

the uplink is given as

max∑

i,j

λi,jσ2 (6)

s.t. Λi,j ≥ γi,j , i = 1, . . . , N, j = 1, . . . ,K

where the uplink SINR, Λi,j is given by

Λi,j =λi,j |wH

i,jhi,i,j |2∑

(m,l) 6=(i,j) λm,l|wHi,jhi,m,l|2 + αi||wi,j ||22

10

where w denotes the corresponding uplink beamforming vector and λi,j repre-

sents the dual variable associated with the optimization problem in (5). The

λi,j can be viewed as the dual uplink power.

Before stating the optimal beamforming algorithm, we define the follow-

ing matrices. Let Hi = [hi,1,1hi,1,2 · · ·hi,m,n · · ·hi,N,K ] be the matrix whose

columns are formed by the channel vectors from BS i to all the UTs across all

the cells. Similarly, Λ = diag[λ1,1λ1,2 · · ·λm,n · · ·λN,K ] be a diagonal matrix

with diagonal elements being the uplink power allocations. We also define the

matrix Σi as

Σi = HiΛHHi (7)

We define the Stieltjes transform ([6]) of the empirical eigen value distribution

of the matrix Σi by the notation mΣi. When the number of antennas Nt on

the MBS and the number of UTs per cell K, become very large, (Nt,K → ∞)

5

• The iterative function for evaluating the optimal uplink power allocation

λi,j is given by

λi,j =(

(

1 +1

γi,j)( σ2

i,i,jmΣi(−αi)

1 + σ2i,i,jλi,jmΣi

(−αi)

)

)−1

(8)

• The optimal receive uplink beamformers can be evaluated as

wi,j =(

∑

m,l

λm,lhi,m,lhHi,m,l + αiI

)−1

hi,i,j (9)

• The optimal transmit downlink beamformers are computed using

wi,j =√

δi,jwi,j (10)

5The details of the algorithm can be found in [5].

11

The parameter δi,j is a scaling factor between the uplink and the downlink

beamforming vectors. It can be computed as in [5].

The effectiveness of our algorithm lies in the fact that the computation of

the uplink power parameter λ depends only on the channel statistics and not

upon the actual channel realization. Hence in order to compute the beamform-

ing vectors, MBSs only need to exchange the channel statistics (of their UTs)

between themselves. This tremendously reduces the computational complexity

and the information to be exchanged between between the BSs especially in a

fast fading scenario in which the channel realization changes rapidly where as

the channel statistics remain constant over fairly long periods of time (assuming

low mobility scenario).

Note however, that the algorithm is optimal in the asymptotic setting im-

plying that the SINR constraints are perfectly satisfied only when the number

of antennas on the MBS and the number of UTs tend to infinity. However, in

a practical scenario, when the network dimensions are finite, the authors in [5]

show with the help of simulation results that the actual achieved SINR fluctu-

ates around the target SINR. The fluctuations around the target SINR depends

upon the channel realizations across the multiple coherence intervals.

3.1.1. The Problem of Feasibility

We now include a practical constraint in the beamformer design problem

namely the max-power constraint on each MBS. We rewrite the optimization

problem of Equation (5) incorporating the max-power constraint.

minwi,j

∑

i,j

wHi,jwi,j (11)

s.t. Γi,j ≥ γi,j , i = 1 . . .N, j = 1 . . .K∑

j

wHi,jwi,j ≤ Pmax i = 1 . . .N

The optimization problem in (5) has a solution only if the max-power constraints

for each MBS is satisfied. In other words, if the problem is infeasible, it implies

that given the max-power constraints on the MBSs, the targeted SINR lie out-

side the capacity region of the system. In this work, we address this problem

12

by selectively reducing the target SINR of a few UTs. We provide a heuristic

approach to select the UTs whose target SINRs can be reduced. For the cells

in which∑

j wHi,jwi,j ≥ Pmax, do the following steps.

1. Denote Pi,j as the downlink power for the UTi,j given by Pi,j = wHi,jwi,j .

Among the UTs belonging to the cells for which power constraint is vi-

olated, select the set of UT(s) which has the maximum ratio δi,j =Pi,j

qi,j.

We define the UTi,j′ = argmaxj δi,j .

2. If the target SINR γi,j′ > τ, decrease the target SINR for UTi,j′ by a

small amount ∆. Else, choose the UT(s) for which the parameter δi,j′ is

the next highest.

3. Solve the optimization problem in Equation (5) with the new set of target

SINR constraints.

4. Repeat the steps 1− 4 until the problem becomes feasible.

The intuition behind the algorithm is as follows. During each time slot t, if the

max-power constraints for the BSs is not met, we start by choosing the set of

UTs for which δi,j is maximum. The numerator term is the power consumed

by the UT. If the UT is consuming a lot of power, then the UT probably has a

bad channel realization and hence we reduce its target SINR. The denominator

term is the queue-length at time slot t. Minimum queue-length implies that

the UT(s) has lesser amount of information bits to be served and hence we

can afford to reduce the data rate of that UT. The parameter ∆ is a design

parameter which can be chosen suitably depending on the simulation scenario.

The parameter τ is a small quantity which ensures that each UT gets at least a

minimum rate. No UT goes unserved.

Therefore, the actual achieved SINR in the downlink fluctuates around the

target SINR both due to the asymptotic nature of the decentralized beamformer

design algorithm and also the feasibility issue. These fluctuations of the SINR

result in a variation of queue-length around a given target queue-length. More-

over, the fluctuations depend on the actual channel realization during each time

slot of which the CS is oblivious. Hence, we need an effective queue-length con-

13

trol algorithm which accounts for the fluctuations of the SINR. We will defer

the description of the H∞ based flow controller to Section 4.

3.2. Beamformer Design and Power Allocation: The Weighted Sum-rate Maxi-

mization Case

We now take up the weighted sum rate maximization case in (3). As in

the case of downlink power minimization problem discussed in the previous

subsection, we decompose the problem in (3) into two parts namely the sum-

rate maximization and queue-length stabilization. The problem of weighted

sum rate maximization is given by

maxwt

i,j

∑

i,j

βi,j log(1 + Γti,j), ∀t = 1 . . . T (12)

s.t.∑

j

wtH

j wti,j ≤ Pmax, t = 1 . . . T, i = 1 . . .N

Optimizing over the beamforming vector in the case of MIMO downlink channel

is a particularly challenging problem even in the centralized scenario (MIMO

broadcast channel) which has been addressed in the past literature [18],[19].

Noting that the main objective of the paper is to bring out the effectiveness

of the H-Infinity queue length controller, we avoid the complications associated

with the above optimization problem by fixing the direction of the beamforming

vector and optimize only over the power allocation associated with the UTs.

We choose the MMSE beamforming vector to fix the direction of beamforming

vector (di,j) given by

di,j =

(

∑

m,l hi,m,lhHi,m,l + I

)−1

hi,i,j

||(

∑

m,l hi,m,lhHi,m,l + αiI

)−1

hi,i,j ||2

(13)

and the beamforming vector wi,j is given by

wi,j = Pi,jdi,j

where Pi,j is the power allocation to the UTi,j . As shown in the iterative algo-

rithm in [18],[19], the MMSE beamforming direction is a reasonable choice for

14

optimizing the weighted sum rate. Hence the modified optimization problem is

given by

maxP t

i,j

∑

i,j

βi,j log

(

1 +P ti,j |d

tHi,jh

ti,i,j |

2

∑

l 6=j Pti,l|d

tHi,lh

ti,i,j |

2 +∑

m 6=i,n Ptm,n|d

tHm,nh

tm,i,j |

2 + σ2

)

, ∀t(14)

s.t.∑

j

P ti,j ≤ Pmax, t = 1 . . . T, i = 1 . . .N

Once again, as in the previous case, we assume that the BSs are only allowed

exchange the channel statistics and not the instantaneous CSI values. Hence, the

MBSs optimize the power allocation based on the asymptotic approximations

of the associated quantities which are given by

|dtH

i,jhti,i,j |

2 −

(

σ2i,i,jmΣi

1 + σ2i,i,jmΣi

)2

a.s.−−−−−−→Nt,K→∞

0 (15)

|dtH

i,lhti,i,j |

2 −(σ2

i,i,jσ2i,i,l/Nt)m

′Σi

(

1 + σ2i,i,jmΣi

)2 (

1 + σ2i,i,lmΣi

)2

a.s.−−−−−−→Nt,K→∞

0 (16)

|dtH

m,nhtm,i,j |

2 −(σ2

m,i,jσ2m,m,n/Nt)m

′Σm

(

1 + σ2m,i,jmΣm

)2 (

1 + σ2m,m,nmΣm

)2

a.s.−−−−−−→Nt,K→∞

0 (17)

Here mΣiis the Stieltjes transform of the matrix Σi =

(

∑

m,l hi,m,lhHi,m,l + I

)

.

The reader can refer to [5] for the derivations using random matrix theory. The

details have been omitted in this paper. In order to simplify notations, let us

introduce

Gi,i,j△

=σ2i,i,j

(

1 + σ2i,i,jmΣi

)2 Gi,i,l△

=σ2i,i,l

(

1 + σ2i,i,lmΣi

)2

Gm,i,j△

=σ2m,i,j

(

1 + σ2m,i,jmΣm

)2 Gm,m,n△

=σ2m,m,n

(

1 + σ2m,m,nmΣm

)2

Let us denote the downlink SINR obtained by replacing the numerator and

denominator terms by their asymptotic equivalents by Γi,j ,

Γi,j =Pi,jσ

2i,i,jGi,i,jmΣi

∑

l 6=j Pi,lGi,i,jGi,i,lm′Σi

+∑

m 6=i,n Pm,nGm,i,jGm,m,nm′Σm

+ σ2(18)

15

Note that with the asymptotic approximation, the downlink SINR Γi,j is the

same across the time slots (since Γi,j only depends upon the channel statistics

which is constant across time slots). We re frame the problem with asymptotic

approximations given by

RasympMAX : max

Pi,j

∑

i,j

βi,j log(

1 + Γi,j

)

(19)

s.t.∑

j

Pi,j ≤ Pmax, i = 1 . . .N

The MBSs can now solve the optimization problem RasympMAX by exchanging only

the channel statistics between themselves. We make the high SINR approxima-

tion and use log(1+x) ≈ log(x), for large x. However, the optimization problem

in log(Γi,j) is not concave in terms of P. Hence, we proceed by using geometric

programming approach [20] and make the variable change to Pi,j = log(Pi,j).

RasympMAX : max

Pi,j

log(

exp(Pi,j)σ2i,i,jGi,i,jmΣi

)

(20)

− log

∑

l 6=j

exp(Pi,l)Gi,i,jGi,i,lm′Σi

+∑

m 6=i,n

exp(Pm,n)Gm,i,jGm,m,nm′Σm

+ σ2

s.t.∑

j

exp(Pi,j) ≤ Pmax, i = 1 . . .N

The problem is now a convex optimization problem in terms of the variable

Pi,j (refer to [20] for details). We Introduce the Lagrange variable δi for the

max-power constraint and formulate the Lagrangian as

L(P, δ) =∑

i,j

βi,j log(

exp(Pi,j)σ2i,i,jGi,i,jmΣi

)

(21)

− log

∑

l 6=j

exp(Pi,l)Gi,i,jGi,i,lm′Σi

+∑

m 6=i,n

exp(Pm,n)Gm,i,jGm,m,nm′Σm

+ σ2

(22)

−∑

i

δi

∑

i,j

exp(Pi,j)− Pmax

(23)

Let us also denote the terms inside log in equation (22) by the notation Ii,j . We

also define P =[

P1,1, P1,2, . . . , P1,K , . . . , PN,K

]

. The solution to the optimiza-

16

tion problem can now be given by

minδ

maxP

L(P, δ) (24)

We will use the gradient based method to solve the dual problem [21]. At the

iteration count τ for the gradient based algorithm, for a given δ = δ(τ), let us

define

P(τ) = argmaxP

L(P, δ(τ)) (25)

The update equation for the dual variable is given by

δi(τ + 1) = δi(τ) − α

∑

j

exp(Pi,j(τ)) − Pmax

∀i = 1, . . . , N (26)

In order to solve the optimization problem in (25), we once again use the gradient

based method. The gradient of L(P, δ(τ)) with respect to Pi,j at iteration ζ is

given by

∆i,j(ζ, τ) =∂L(P, δ(τ))

∂Pi,j

∣

∣

∣

Pi,j=Pi,j(ζ,τ)

In particular, we have

∆i,j(ζ, τ) = βi,j −∑

l 6=j

βi,l exp(Pi,l(ζ, τ))Gi,i,lGi,i,jm′Σi

Ii,l

−∑

m 6=i,n

βm,n exp(Pm,n(ζ, τ))Gi,m,nGi,i,jm′Σm

Im,n

− δi(τ) exp(Pi,j(ζ, τ))

We iteratively update the power allocation until convergence

Pi,j(ζ + 1, τ) = Pi,j(ζ, τ) + κ∆i,j(ζ, τ) i = 1, . . . , N, j = 1, . . . ,K

where κ is a small step size. The value Pi,j(τ) is the value of the gradient

algorithm at the convergence point given by,

Pi,j(τ) = limζ→∞

Pi,j(ζ, τ)

17

Now, let use denote the solution of the optimization problem RasympMAX as

µi,j = maxPi,j

RasympMAX

Pi,j = argmaxPi,j

RasympMAX

The MBSs precompute the power allocation based on the channel statistics to

Pi,j and assume a precomputed service rate µi,j . The precomputed service rate

can be viewed as an estimate of the service rate offered to the queues at the CS.

However, in reality, the actual service rate offered to the queues depends on the

channel realization and fluctuates around the target service rate. The service

rate obtained per time slot is given by

µti,j = log

(

1 +Pi,j |d

tHi,jh

ti,i,j |

2

∑

l 6=j Pi,l|dtHi,lh

ti,i,j |

2 +∑

m 6=i,n Pm,n|dtHm,nh

tm,i,j |

2 + σ2

)

The CS has only an estimate of the service rate provided to the UTs given by

µi,j . However, the CS being oblivious to the actual channel realization does not

have the knowledge of the actual service rates in the downlink.

Hence, it is clear that in both the optimization problems (downlink power

minimization and weighted sum rate maximization) there is a mismatch between

the estimated service rate and actual service rate obtained to the queues. We

model the mismatch between the targeted service rate and the actual service

rate as unknown noise and apply H∞ based queue length controller described

in section 4 to regulate the arrival rates into the queues.

In what follows, we provide a flow control algorithm based on H∞ control.

4. Scheduler Design and Queue-Length Stability at the MBS

We now describe the scheduler design. The scheduler has to regulate the

flow into the queues without the knowledge of the wireless channel conditions.

(which implies that the scheduler does not have the knowledge of the actual

received SINR in the downlink, µti,j).

18

Let us now focus on the queue length dynamics at the MBS. The equation

for the queue-length evolution at the MBS is given by

qt+1i,j =

(

qti,j + ati,j − µti,j

)+i = 1 . . .N, j = 1 . . .K (27)

It must be recalled that we assumed the CS to be an infinte reservoir of packets.

The flow controller at the CS (to be described later in this section) specifies ati,j ,

the number of packets that should flow into the queue at the MBS from the CS

at each time slot t and µti,j is the number of packets departing the queue (based

on the downlink SINR corresponding to the beamforming vector design). Let

us subtract the target queue-length qi,j from the LHS and the RHS of (27). We

also add and subtract the target service rate of each queue, µi,j on the RHS of

the equation (27). Hence equation (27) after rearranging becomes

qt+1i,j − qi,j =

(

[qti,j − qi,j ] + [ati,j − µi,j ] + [µi,j − µti,j ])+

(28)

We perform a change of variables and define ψti,j = qti,j − qi,j and the vectorized

version by the notation ψt = [ψt1,1, ψ

t1,2, . . . , ψ

t1,K , . . . , ψ

tN,K ]T . We also define

uti,j = ati,j−µi,j and its vectorized notation by ut = [ut1,1, ut1,2, . . . , u

t1,K , . . . , u

tN,K]T .

Similarly ζt = (µi,j−µti,j) and its vectorized notation ζt = [ζt1,1, ζ

t1,2, . . . , ζ

t1,K , . . . , ζ

tN,K ]T .

Therefore, Equation (28) in its vectorized form becomes

ψt+1 = ψt + ut + ζt (29)

Note that we have removed the (.)+ operation assuming that the queue-length

remains positive at all time instants. The CS makes an observation on the queue-

length in order to regulate the traffic arrival into the queues. The observation

equation is given by

yt = ψt + ηt (30)

where ηt is a noise parameter. The noise parameter is representative of the

fact that the CS only an estimation of queue-length. Moreover, in practical

scenario this observation may even be an outdated observation. We model the

19

observation errors as noise. Note that we do not assume any particular statistical

distribution for the noise parameter.

The objective of the scheduler design problem is to minimize the cost func-

tion given by

J = limT→∞

1

T

[

T∑

t=1

(

||ψt||2W + ||ut||2Q)]

(31)

where W and Q are weight matrices. In our algorithm, both the matrices are

assumed to be identity matrices. The first term of the cost function in Equation

(31) denotes the penalty for deviating from the target queue length. The second

term tries to maintain the arrival rate at each queue as close as possible to the

target service rate γi,j . Note that the scheduler is unaware of the error term

ζt, since it does not have the knowledge of the actual received SINR in the

downlink.

Equation (32) represents a linear dynamic system with the state variable de-

noted by the notation ψt, a control parameter denoted by the notation ut and

an unknown process noise denoted by ζt. In particular, Equation (32) denotes

the evolution of the state variable and (31) the quadratic cost to be minimized.

We use a controller based on H∞ control algorithm [7] to solve the above prob-

lem. The task of the controller is to specify an arrival rate into the queue by

calculating the control parameter ut at each time slot t.

The reason why H∞ filter makes controlling decisions without making any

assumptions of the noise processes is becauseH∞ control assumes the worst case

noise. That is, it tries to minimize the cost function assuming the worst case

noise. For this reason, the H∞ control is called mini-max control. Additionally,

the scheduler is trying to minimize the long term cost function. The problem

is a dynamic programming problem involving stochastic processes i.e., during

each time slot the CS has to take control decisions such that the long term cost

function is minimized (and not the instantaneous cost function). For details of

the H∞ control design please refer to the Appendix A.

We would like to state that in practice, we can add a scaling factor α as a

weight to the control parameter. We define a matrix S = diag( 1√α. . . 1√

α). The

20

equation defining the evolution of the state variable becomes

ψt+1 = ψt + Sut + ζt (32)

And hence, the correspondingly, the weight matrix for the control variable be-

comes V = diag(α . . . α). The parameter α can be varied to adjust the conver-

gence rate of the H∞ control.

5. LQG Control Vs H-Infinity

In this section, we compare the effectiveness of our H-infinity controller in

minimizing the fluctuations of the queue-length as opposed to a standard LQG

controller. It must be noted that standard LQG control assumes that the process

noise is Gaussian in nature and minimizes the expected value of the cost func-

tion. In general, the nature of the fluctuations of the service rate to the queues

around the target service rate would depend on the fluctuations of the wireless

channel which is unknown to the CS. In this section, we assume a Gaussian ap-

proximation to model the fluctuations and perform simulations with standard

LQG control (which is known to be optimal under Gaussian noise). We provide

a brief description of a LQG control in Appendix B.

The simulation results (refer to section 6, Figure 7) show that the H-infinity

flow controller performs better than the LQG controller in terms of keeping

the fluctuations of the queue-length to minimum. The above observation shows

that the Gaussian approximation is not a very accurate representation of the

process noise. Hence, using a robust controller such as H∞ control keeps the

queue-length fluctuations to a lower value and ensures stability.

6. Numerical Results

In this section we provide some numerical results to show the effectiveness

of our algorithm. We consider a hexagonal cellular system with a cluster of 3

cells as shown in Figure 2. Each cell has a MBS which serves only the UTs in

its cell. The number of transmit antennas on each MBS scales with the number

21

of UTs in the cell such that their ratio is a finite constant. In particular, we

assume Nt = K. The MBS are connected to the CS. We assume that the target

queue-length of the buffers at all the UTs to be 20kBs. We also assume the

number of channel uses per coherence interval of the channel (B in equation

(4)) as one-third the target queue-length. The channel realizations are assumed

to be i.i.d. across the coherence intervals.

BS1

BS2

BS3

d1,1,1

d2,1,1

d3,1,1

Figure 2: Example of a network with 3 Cells. The crosses represent the location of the BSs and

the dots represent the location of the UTs randomly scattered inside the cells. The distances

of a UT from the three BSs are also provided.

We consider a distance dependent path loss model in which the UTs are

assumed to be arbitrarily scattered inside each cell. In this case, the path loss

factor from MBS i to UTj,k is given by σ2i,j,k =

(

1di,j,k

)β

where di,j,k is the

distance between MBS i to UTj,k, normalized to the maximum distance within

a cell, and β is the path loss exponent which lies usually in the range from 2

to 5 dependent on the radio environment. We normalize the variance of the

total received noise to σ2 = 1. We also assume that no user terminal is within

a normalized distance of 0.1 from the closest BS.

We perform our simulations for the case of downlink power minimization

(Section 3.1) and assume a target SINR of 9 dB for all the UTs during each

time slot. First let us focus on the case with no max-power constraint on the

22

MBSs (Pmax = ∞). We run our simulations for T = 100 time slots. We first

plot the variation of queue-length values (averaged across the UTs) against the

number of UTs per cell in Figure 3. The bubbles represent queue-length values

at each time slot for a given number of UTs (overlapped on the same point) and

the horizontal line represents the target queue-length.

In order to quantify the variation in the queue-lengths around the target

queue size, we define the Normalized Mean Square Error (NMSE) of the queue-

lengths given by

NMSE =1

NKT

T∑

t=1

∑

i,j

(qti,j − qi,j)2

qi,j

The NMSE for the arrival rates also follows a similar definition. We plot the

NMSE of the queue-lengths against the number of UTs in Figure 4.

It can be seen that in both the Figures 3 and 4 (in this case the curve

corresponding to Pmax = ∞), as the number of UTs increases, the variance of

the queue-length becomes lesser. This is in accordance with our beamforming

algorithm. Recall that as the number of UTs become large, the achieved SINR

in the downlink gets very close to the target SINR for every given channel

realization. Hence, the variation of the queue-length around the target also

reduces.

We also plot the NMSE of the arrival rates into the queues specified by the

scheduler in Figure 5. The NMSE of the arrival rates also reduce as the number

of UTs increase. In fact, when the number of UTs goes to infinity, effectively,

we do not require queue-length control anymore. The arrivals into the queue is

then deterministic equal the the target queue-length at all the time slots.

We now introduce the max-power constraint on each MBS. When the max

power constraint of a particular MBS is not satisfied, we use the approach

developed in Section 3.1.1 to redesign the system parameters. In Figure 4 and

5 (curves corresponding to Pmax = 30, 40), we plot the NMSE of the queue-

lengths and the arrival rates with power constraints. Please note that Pmax

indicated in the plots is normalized by K, the number of UTs. It can be seen

that as the Pmax per MBS reduces, the NMSE parameter increases. This goes

23

well with the intuition since with the max-power constraint, the error between

the target SINR and actual achieved SINR in the downlink increases because

of redesigning of system parameters. The max-power plots will be useful in

determining the excess buffer capacity beyond the target queue-length that the

system designer will have to provide.

0 10 20 30 40 500

0.5

1

1.5

2

2.5

3x 104

No. of UTs per Cell

Que

ue−

Leng

ths

Target Queue−Length

Figure 3: Scatter Plot showing the variation of queue-length around the target, Target SINR

= 9dB, Target Buffer Length = 20kBs, β = 3.6, T =100

5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

3000

No. of UTs per Cell

NM

SE

Que

ue−

Leng

th

Pmax

= 30

Pmax

= 40

Pmax

= ∞

Figure 4: NMSE of Queue-Length Vs The No. of UTs. Target SINR = 9dB, Target Buffer

Length = 20kBs, β = 3.6, T = 100

In Figure 6, we plot the average downlink transmitted power per UT and the

24

5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

No. of UTs per Cell

NM

SE

− A

rriv

al R

ate

Pmax

= 30

Pmax

= 40

Pmax

= ∞

Figure 5: NMSE of Arrival Rates Vs The No. of UTs. Target SINR = 9dB, Target Buffer

Length = 20kBs, β = 3.6, T = 100

10 20 30 40 50 60 701

1.5

2x 104

Max−Power Constraint

Avg

. Rat

e A

chie

ved

per

UT

10 20 30 40 50 60 700

50

100

Avg

. Pow

er C

onsu

med

per

UT

Figure 6: Mean Achieved Rate per UT (left y-axis) and Mean Power Consumed (Right y-

axis) per UT Vs Max-Power of the BSs. Target SINR = 9dB, Target Buffer Length = 20kBs,

β = 3.6, T =100, 10 UTs per cell

25

average rate achieved per UT against the given max-power constraint (Pmax) in

the same plot with 10 UTs per cell and 100 channel realizations. It can be seen

that more the power consumed, more is the average rate achieved per UT . This

implies that with more power, the MBSs can meet the targeted SINR without

having to reset them to lower values. Hence, the error between requested SINR

(target) and actual achieved SINR is lower. Also, for low levels of Pmax, it can

be seen that the increase in total downlink transmitted power is almost linear

as a function of Pmax. This is due to the fact that at low Pmax, all the MBSs are

transmitting nearly at their peak in order to provide the best possible data rates

for the UTs (with the power constraint). However, as Pmax increases beyond

certain value, the system performance is close to the case of Pmax = ∞. This

is due to the fact that with higher powers, the targeted SINRs lie inside the

capacity region.

10 20 30 40 500

200

400

600

800

1,000

1,200

No. of UTs per cell

NMSE

Queue-Length

LQG Control

H-Infinity Control

Figure 7: Comparison of H∞ control Vs LQG control, Target SINR = 9dB, Target Buffer

Length = 20kBs, β = 3.6, T = 100

We finally plot the comparison of H∞ control in comparison with LQG

control for the flow controller in Figure 7. The H-infinity controller provides

better performance in terms of minimizing the fluctuations of the queue-length

26

as discusses in Section 5.

7. Conclusions

In this work, we addressed the joint problem of traffic scheduling using H∞

control theory and decentralized beamforming design using tools from random

matrix theory in the context of cellular networks. The task of the H∞ based

flow controller is to regulate the arrival process to the queues at the MBS from

the CS. The H∞ based flow controller stabilizes the queue-lengths at the MBS

without the knowledge of the wireless channel between the MBS and the UTs.

The H∞ controller keeps the fluctuations around the target queue-length to a

minimum as compared to LQG based control.

In this work we decoupled the two problems of traffic scheduling and beam-

forming design. A natural extension of this work would be to jointly address the

two problems of flow control and beamforming design and develop a cross-layer

model to address the same.

Appendix A: H-Infinity Control

In this section, we provide a brief overview on the solution to the discrete-

time system dynamic with unpredictable noise. (Please note that in order to

maintain consistency with the previous literature in this field, we deviate from

the guidelines set for notations in the introduction section this paper). We

consider the following state space equation:

xt+1 = Axt +But +D1Γt (33)

where state vector X, control vector U and noise vector Γ take values respec-

tively in Rn, Rp and Rm. Moreover, we define the following measurement equa-

tion for the state (i.e. the state is not known exactly but estimated via an

observation),

yt = Cxt +D2Γt (34)

27

observable state vector Y takes values in Rn. The objective is to design a

controller that minimizes the following cost function:

L = limT→∞

{ 1

T

T∑

t=0

(

||xt||2W + ||ut||2Q

)}

(35)

where W and Q are the norm matrices. The problem to solve is then a linear

control problem with quadratic cost and unpredictable noise. Please note that

when the noise is Gaussian, the solution to this problem is to use LQG (Linear

Quadratic Gaussian) technique. When the noise is unpredictable (we don’t

have information on the distribution of the noise), an efficient way to solve the

aforementioned problem is to use H∞ technique. The H∞ controller adopted

in this section is the robust controller discussed in [7] and based on zero sum

game theory. To solve our problem using H∞, we introduce the following cost

function:

Jπ = limT→∞

{ 1

T

T∑

t=0

(

||xt||2w + ||ut||2Q − π2||Γt||2)}

(36)

where, π is the level of attenuation. This is a minimax optimization problem

where the cost function is minimized over maximum unknown disturbance. This

problem is a zero sum game with two players. The cost Jπ is minimized by player

1 and maximized by player 2 using vectors U and Γ. One can refer to [7] for more

general class of discrete-time zero-sum games, with various information patterns,

where sufficient conditions for the existence of a saddle point are provided when

the information pattern is perfect and imperfect state.

The H∞ controller can be obtained according to the following proposition

[7];

There exists a state feedback H∞ controller Ut such that,

ut = M(I+ (1−1

π2)Σ)−1x

t (37)

where, ρ(ΣM) < π2. Moreover,

1) M is a minimal non-negative definite solution to the following algebraic

Riccati equation,

M = I+M(I+ (1−1

π2)M)−1 (38)

28

where,

π2I−M > 0 (39)

2) Σ is a minimal non-negative definite solution to the following algebraic Riccati

equation,

Σ = I+Σ(I+ (1−1

π2)Σ)−1 (40)

where,

π2I− Σ > 0 (41)

3) and the state estimate Xtis generated by,

xt = (I+Σ− π−2ΣM)−1(xt +Σyt) (42)

and

xt+1 = xt − ut +Σ(I+ (1−1

π2)Σ)−1(yt − (1−

1

π2)xt) (43)

The above proposition allows us to find the optimal controller for given π.

Therefore, to obtain the solution, we have to find the optimal value of π2 which

satisfies all the above conditions. For that, we use the following lemma: We use

the following algorithm to find the optimal π [ [12], [7], [22] [23] ].

1. Start with a small value of π2 ≥ 0.

2. Increment the value of π2 by a small step size δ.

3. Formulate the two Hamiltonian matrices corresponding to the Ricatti

Equations

H1 = H2 =

I −(

1− 1π2

)

I

−I −I

For our particular problem, the two Hamiltonian matrices H1 and H2 are

identical.

4. Find the eigen values of H1 and H2.

5. If the eigen value of H1 or H2 has an imaginary part, then go to step 2.

ELSE for this value of π2 solve the Ricatti Equations in (38) and (40) to

find the values of M and Σ.

6. If M < 0 or Σ < 0, go to step 2 ELSE check conditions (39) and (41).

29

7. If conditions (39) and (41) are not satisfied, go to step 2. Else check the

spectral radius of ΣM.

8. If ρ(ΣM) < π2, go to step 2, ELSE π2 is the optimal value and terminate

the algorithm.

Appendix B: LQG Control

We now provide a brief description of standard LQG control algorithm. Con-

sider the following state space equation and the observation equations given by

xt+1 = Axt +But +wt

yt = Cxt + vt

where xt is the state variable, ut the control and wt and vt being the process

and the observation noise respectively. The sequences vt and wt are Gaussian

distributed with zero mean and covariance matrix given by

E[

wtT ws]

= R1δt,s

E[

wtT vs]

= R2δt,s

E[

wtT vs]

= 0

where δt,s are Dirac functions. The cost function to be minimized is given by

J = limT→∞

1

TE

[

T−1∑

t=0

||xt||2W + ||ut||2Q

]

The LQG control with imperfect state observations is summarized as follows

[24]. Starting with the initial state estimate xt∣

∣

t=0= 0, solve the following

Ricatti equations given by

S = ATSA−ATSB(

BTSB+Q)−1

BTSA+W

P = A(P −PC(CPCT +R2)−1CP)AT +R1

The optimal control strategy which is a linear control law in the state estimate

is given by

ut = −Lxt

30

where the matrix L is given by

L =(

BTSB+Q)−1

BTSA

The state estimate at time t+ 1 is given as a function of the state estimate at

t and observation yt is given according to the following equation

xt+1 = Axt +But +K(yt −Cxt)

where the matrix K is given as

K = APCT (CPCT +R2)−1

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